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Fundamental Theorem of Calculus

Fundamental Theorem of Calculus. Finally!. Objective…. To integrate using the Fundamental Thm of Calc. Pandora’s box…. Fundamental Thms. The Fundamental Theorem of Arithmetic: Any positive integer can be represented in exactly one way as a product of primes.

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Fundamental Theorem of Calculus

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  1. Fundamental Theorem of Calculus Finally!

  2. Objective… • To integrate using the Fundamental Thm of Calc

  3. Pandora’s box…

  4. Fundamental Thms • The Fundamental Theorem of Arithmetic: • Any positive integer can be represented in exactly one way as a product of primes. • The Fundamental Theorem of Algebra: • Every polynomial of degree n has exactly n zeroes. • The Fundamental Theorem of Geometry: • No theorem wears this title, but perhaps the Pythagorean Theorem deserves it.

  5. Integrals… area under the curve • No problem if it’s a geometric shape… (4.3) • What if it’s not? How could we find the area under the curve?

  6. Rectangles…

  7. An easier example…. • This is called Riemann Sums • Using left-hand endpoints with 4 rectangles • Area =

  8. What if…. • We use right-hand endpoints and 4 rectangles? • Area =

  9. What’s a more accurate way to find area?

  10. How many rectangles is the best? f(x) = y- value or height and Δx = (b-a)/n (n is the number of rectangles)

  11. Riemann Sums and definite integrals

  12. Fundamental Theorem of Calculus • If f is cont on [a,b] and F is an antiderivative of f on [a,b] then

  13. Example

  14. What about a + C?

  15. Absolute values…

  16. A different example • Find the area of the region bounded by y=2x^2 – 3x + 2, x-axis, x = 0, and x = 2. • Step 1… draw graph

  17. Ex cont… • Find the area of the region bounded by y=2x^2 – 3x + 2, x-axis, x = 0, and x = 2. • Step 2: Write the integral and integrate

  18. Average Value of a function • Average value = Find the average value of f(x) = 3x^2 – 2x on [1,4]

  19. Pg 283, #31 • A company purchases a new machine for which the rate of depreciation is dV/dt = 10,000(t-6) where 0< t< 5 and V is the value of the machine after t years. What is the total loss of value of the machine over the first 3 years?

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